Does the Schrodinger Equation care about spin? I have taken the non-relativistic limit of the Klein-Gordon and Dirac equation, and both have brought me to the Schrodinger equation.
The Klein-Gordon equation describes spin 0 particles, and the Dirac equation describes spin $\frac12$ particles, and yet in the non-relativistic case they are equal.
I'm starting to wonder if spin is a relativistic thing, but I know that I studied it in Griffiths' Introduction to Quantum Mechanics, which is all non-relativistic.
Why is it that the Schrodinger equation can describe particles of seemingly any spin, and in the relativistic case we have to be so careful about it?
 A: There are some misconceptions in what you wrote. 
There are four (sets of) equations, all discovered in 1926-1928. The Schrödinger equation (1926, spin $0$, non-specially relativistic), the Pauli equation(s) (1927, spin $1/2$, non-specially relativistic), the Klein-Gordon equation (1926, spin $0$, specially relativistic), the Dirac equation(s) (1928, spin $1/2$, specially relativistic). So you can now understand how they are related, which simplification/generalization you need to do, in order to move from one to another. 
Spin is a relativistic thing, either Galilean relativistic, or specially-relativistic. 
If you say "Why is it that the Schrödinger equation can describe particles of seemingly any spin?", then this is true, only if the Schrödinger equation is cast into the form written in ACuriousMind's comment, that is the Hamiltonian is not defined to be the one taken by Schrödinger in 1926, but can be any of them, including the Dirac Hamiltonian. 
A: The Schrödinger equation is the nonrelativistic limit of the Klein-Gordon - not of the Dirac - equation, hence it does not contain spin. You can add spin by interpretating the Schrödinger wave function as a sponsor and adding the Pauli spin interaction. This equation is the nonrelativistic limit of the squared Dirac equation. In Itzykson&Zuber the Dirac equation is solved for hydrogenic atoms via the squared Dirac equation. Highly recommended. 
